1. **Problem Statement:** We have two classes with their test scores: Class A: 2, 7, 5, 6, 9, 10, 2, 2, 3, 10 Class B: 4, 5, 6, 4, 4, 5, 6, 7, 4, 6 We need to find for each class: a) Mean, median, and mode b) Variance and standard deviation c) Which class has the highest variability 2. **Formulas and Rules:** - Mean: $\bar{x} = \frac{\sum x_i}{n}$ where $n$ is number of scores. - Median: Middle value when data is sorted. - Mode: Most frequent value. - Variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ (sample variance) - Standard deviation: $s = \sqrt{s^2}$ 3. **Calculations for Class A:** - Sorted scores: 2, 2, 2, 3, 5, 6, 7, 9, 10, 10 - Mean: $\frac{2+7+5+6+9+10+2+2+3+10}{10} = \frac{56}{10} = 5.6$ - Median: Middle two values are 5 and 6, so median = $\frac{5+6}{2} = 5.5$ - Mode: 2 appears 3 times, more than others, so mode = 2 4. **Variance and Standard Deviation for Class A:** Calculate squared deviations: $(2-5.6)^2=12.96$, $(7-5.6)^2=1.96$, $(5-5.6)^2=0.36$, $(6-5.6)^2=0.16$, $(9-5.6)^2=11.56$, $(10-5.6)^2=19.36$, $(2-5.6)^2=12.96$, $(2-5.6)^2=12.96$, $(3-5.6)^2=6.76$, $(10-5.6)^2=19.36$ Sum = 98.04 Variance: $s^2 = \frac{98.04}{9} \approx 10.89$ Standard deviation: $s = \sqrt{10.89} \approx 3.30$ 5. **Calculations for Class B:** - Sorted scores: 4, 4, 4, 4, 5, 5, 6, 6, 6, 7 - Mean: $\frac{4+5+6+4+4+5+6+7+4+6}{10} = \frac{51}{10} = 5.1$ - Median: Middle two values are 5 and 5, so median = 5 - Mode: 4 appears 4 times, so mode = 4 6. **Variance and Standard Deviation for Class B:** Squared deviations: $(4-5.1)^2=1.21$, $(5-5.1)^2=0.01$, $(6-5.1)^2=0.81$, $(4-5.1)^2=1.21$, $(4-5.1)^2=1.21$, $(5-5.1)^2=0.01$, $(6-5.1)^2=0.81$, $(7-5.1)^2=3.61$, $(4-5.1)^2=1.21$, $(6-5.1)^2=0.81$ Sum = 10.9 Variance: $s^2 = \frac{10.9}{9} \approx 1.21$ Standard deviation: $s = \sqrt{1.21} = 1.1$ 7. **Conclusion:** Class A has variance $\approx 10.89$ and standard deviation $\approx 3.30$. Class B has variance $\approx 1.21$ and standard deviation $1.1$. Therefore, Class A has the highest variability. **Final answers:** - Class A: Mean = 5.6, Median = 5.5, Mode = 2, Variance $\approx$ 10.89, SD $\approx$ 3.30 - Class B: Mean = 5.1, Median = 5, Mode = 4, Variance $\approx$ 1.21, SD = 1.1 - Highest variability: Class A